Undergrad Is Transverse Strain Zero at the Bolts of a Clamped Circular Membrane?

[Chrono G. Xay]

TL;DR

I’m trying to do some analysis of an acoustic drum at rest. In an earlier thread of mine I was getting some help verifying the equation for axisymmetric surface tension of a circular membrane. The starting point I chose assumed that the transverse strain of the clamping annulus was either zero or considered negligible. This next analysis assumes the opposite.

The annulus (“rim”) clamping the circular membrane over its cylindrical shell has a number of bolts ‘n’ positioned equidistantly around its perimeter. I’m guessing that the amount of transverse strain at those bolts would be where we might reasonably assume to be zero.

When I was first trying to imagine what general shape the rim would take between any two neighboring bolts I thought of a catenary curve. However, as time went on that made less and less sense. As I kept poking around with a graphing calculator, what seemed to make more sense is something probably akin to a sort of… ‘static’ (my word) amplitude modulating function. Here’s an example of what I’m talking about (in Cartesian coordinates): $$y_1(x) = \frac{a^{-1}\left|{0.5\left[a - sin\left({nπx + \frac π 2 }\right)\right]} \right| sin\left({nπx + \frac π 2}\right)-0.5\left(a-1\right)} b$$
Where $a ≥ 2$ and $b > 0$.

I’m guessing that a parametrized graph of this line in cylindrical coordinates would be something like… $r = n, z=y_1(θ)$? Did I use that word correctly?

 

[Chrono G. Xay]

Here’s a screenshot of the graphed function (very slightly altered- LaTex below), where $n = 6$, $a = 3$, and $b = 5$:

$$y_1(x) = \frac{\left|{0.5\left[a - sin\left({nπx + \frac π 2 }\right)\right]} \right| sin\left({nπx + \frac π 2}\right)+\left(a-1\right)} {ab}$$
Where $a ≥ 2$ and $b > 0$.

View attachment 340565

For the sake of clarity: The valleys of the graph indicate the locations of the bolts pulling down on the annulus (“rim”), which pulls the circular membrane taught across the opening of the open-ended cylindrical shell. The peaks then are the transverse strain of the rim.

 

[Chrono G. Xay]

The picture I uploaded is giving me errors. Let me try that again…